The table shows the heights of some basketball players in centimetres. Calculate, to three decimal places, the standard deviation of these heights.
So we’ve got the numbers 180, 181, 183, 185, 179, 184, 175, 188, 183, and 184. Now we’ve got to calculate the standard deviation. And we use this symbol here, 𝜎, to represent the standard deviation. Now hopefully recall that the standard deviation can be summed up by calling it the root mean square deviation from the mean.
So that basically means that we have to calculate the mean of the pieces of data that we’ve got. And then for each one, we work out the difference between the mean and that data value and square that. Then we add all those terms up and divide by the number of terms that we’ve got. That gives us an answer which we call the variance.
If we take the square root of that answer, that’s the standard deviation. So first, we’re gonna calculate the mean. And we’ll call that 𝑥 bar. Then we’re going to square each piece of data’s difference from the mean. Then we’re going to add up all those squares of differences, then divide that sum by the number of pieces of data that we’ve got, and then take the square root of that answer.
Okay, let’s make a little bit of space for us to do some calculations. First off then, let’s calculate the mean value of that data. Well, the first thing we need to work out is how many bits of data have we got. And we’ve got one, two, three, four, five, six, seven, eight, nine, 10. So 𝑛, the number of pieces of data, is equal to 10.
Now we’ve got to add up all the pieces of data and divide by how many pieces of data there were. So the mean is all of those numbers added up divided by 10, which gives us 1822 divided by 10. That means if we laid all of the basketball players down on the floor end to end, they would take up 1822 centimetres in total. And if we divided all of that length equally between the 10 of them, then they’d have 182.2 centimetres each. So the mean height is 182.2 centimetres. And that’s the first step complete.
Now we’ve got to go through and square the difference from the mean for each of our basketball players. Well, for the first one, the difference is 182.2 minus 180. And if we square that, we get an answer of 4.84.
For the second player, the calculation becomes 182.2 minus 181 all squared, which gives us 1.44. Now the third basketball player is 183 centimetres tall, which is taller than the mean. So if I do the mean minus their height, I’m gonna get a negative number. But when I square that negative number, a negative number times a negative number is gonna give me a positive number. So the answer to that one is 0.64. And I need to do that for each of the basketball players in turn. And having done it, I’ve now completed step two.
Now we need to do step three. We’ve got to sum all of those square differences. So if I add all of those numbers together, I get an answer of 117.6. And that’s step three complete.
Now step four is to divide that sum by the number of pieces of data. So there were 10 basketball players. We’ve got to divide that by 10. And as you can see, we’re getting towards our formula for standard deviation.
Now we’ve got the sum of the squares of the differences from the mean divided by 10, which is 11.76. Now that is the variance or 𝜎 squared for our dataset for the heights of the basketball players.
But remember, we want the standard deviation 𝜎. So we need to take the square root of that answer. And according to my calculator, that’s 3.42928564 and then loads of other digits. But let’s look back at the question. It told us to calculate to three decimal places. So we’ve got to round that to three decimal places. So that gives us an answer that the standard deviation is 3.429 to three decimal places.